Interval Extrema for a Pell-to-Ternary Numeration Map
Year: 2026
Type: preprint
Abstract: Every nonnegative integer has a unique greedy representation using Pell numbers as place values. This paper keeps that representation word unchanged but reads it as a base-3 number. The resulting map is the increasing enumeration of the nonnegative ternary integers in which every digit 2 is immediately followed by 0. The main result describes the exact normalized behavior of this map on intervals between consecutive Pell numbers. Let κ = log(3)/log(1 + √2). For every k ≥ 4, the ratio a(n)/n^κ has its unique maximum on [P_k, P_{k+1}) at P_k and its unique minimum at the internal seam 2P_k − 1. The interval [P_3, P_4) is the sole exception to this eventual minimum law. The proof introduces a right-supporting-line principle for complete Pell blocks and uses it to establish the unique interval extrema. Exact formulas are also obtained for the boundary and seam values, together with the global liminf and limsup. The greedy Pell representation, its legal digit language, and cross-base evaluation as a general construction are established material. The contribution of this paper is the supporting-line lemma, the exact Pell-interval extrema theorem, and its asymptotic consequences. The archive includes the manuscript, source files, bibliography, reproducibility programs, theorem-development record, OEIS submission draft, release documentation, and SHA-256 checksums.
Author: Foth, Jake
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208836120
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